Problem: The graph of a sinusoidal function has a minimum point at $(0,2)$ and then has a maximum point at $(3\pi,6)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Solution: The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline passes exactly between the minimum value $2$ and the maximum value $6$, so the midline equation is $y={4}$. The extremum points are $2$ units above or below the midline, so the amplitude is ${2}$. The maximum point is $3\pi$ units to the right of the minimum point, so the period is $2\cdot 3\pi={6\pi}$. [Why did we multiply by 2?] Determining the type of function to use Since the graph has an extremum point at $x=0$, we should use the cosine function and not the sine function. This means there's no horizontal shift, so the function is of the form $a\cos(bx)+d$. [How do we know that?] Determining the parameters in $a\cos(bx)+d$ Since the extremum point at $x=0$ is a minimum point, we know that $a<0$. [How do we know that?] The amplitude is ${2}$, so $|a|={2}$. Since $a<0$, we can conclude that $a=-2$. The midline is $y={4}$, so $d=4$. The period is ${6\pi}$, so $b=\dfrac{2\pi}{{6\pi}}=\dfrac{1}{3}$. The answer $f(x)=-2\cos\left(\dfrac{1}{3}x\right)+4$